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SageMath
E = EllipticCurve("fs1")
E.isogeny_class()
Elliptic curves in class 193200fs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.hi4 | 193200fs1 | \([0, 1, 0, -25508, 92988]\) | \(458891455696/264449745\) | \(1057798980000000\) | \([2]\) | \(737280\) | \(1.5725\) | \(\Gamma_0(N)\)-optimal |
| 193200.hi2 | 193200fs2 | \([0, 1, 0, -290008, 59869988]\) | \(168591300897604/472410225\) | \(7558563600000000\) | \([2, 2]\) | \(1474560\) | \(1.9191\) | |
| 193200.hi1 | 193200fs3 | \([0, 1, 0, -4637008, 3841759988]\) | \(344577854816148242/2716875\) | \(86940000000000\) | \([2]\) | \(2949120\) | \(2.2656\) | |
| 193200.hi3 | 193200fs4 | \([0, 1, 0, -175008, 107939988]\) | \(-18524646126002/146738831715\) | \(-4695642614880000000\) | \([4]\) | \(2949120\) | \(2.2656\) |
Rank
sage: E.rank()
The elliptic curves in class 193200fs have rank \(1\).
Complex multiplication
The elliptic curves in class 193200fs do not have complex multiplication.Modular form 193200.2.a.fs
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
